On Error Estimate of an Approximate Method to Solve an Inverse Problem for a Semi-Linear Differential Equation

An inverse problem for a semi-linear differential-operator equation in a Hilbert space is considered in the paper. The projection regularization method is used to get a stable approximate solution to the nonlinear ill-posed problem. The regularization parameter is chosen referring to the Lavrentev scheme. A sharp error estimate of the considered method on a correctness class defined by means of a nonlinear operator is obtained. The value of the continuity module for the corresponding problem on the correctness classes plays an important role in the investigation of the methods for the solution of ill-posed problems in order to state their optimality. The linear operators are used, as a rule, to define the correctness classes. The two-sided estimate of the continuity module for the nonlinear inverse problem on the correctness class defined by a nonlinear operator is obtained in the present work. The obtained estimate of the continuity module is used to prove the order-optimality of the projection regularization method on the analyzed correctness class.